Abstract [en]

Suppose there is a nonnegative function u and an open set Ohm subset of R-n(n greater than or equal to 3), satisfying Deltau = chi (Ohm) in B-r(e), u = \delu\ = 0 on B-r(e)\Ohm, where B-r(e) = {x : \x\ > r}. Under a certain thickness condition on R-n\Ohm, we prove that the boundary of {x:x/\x\(2) is an element of Ohm} is a graph of a C-1 function in a neighborhood of the origin. As a by-product of the method of the proof, we also obtain the following result: Replace chi (Ohm) by f chi (Ohm), with a certain assumptions on f. Then for any solution u which is asymptotically nonnegative at infinity, there holds lim(r-->infinity) (\Br\)/(\Ohm boolean AND Br\) is an element of {1/2,1}.